Optimal control for traffic flow networks

We consider traffic flow models for road networks where the flow is controlled at the nodes of the network. For the analytical and numerical optimization of the control, the knowledge of the gradient of the objective functional is useful. The adjoint calculus introduced below determines the gradient in two ways. We derive the adjoint equations for the continuous traffic flow network model and derive also the adjoint equations for a discretized model. Numerical examples for the solution of problems of optimal control for traffic flow networks are presented.